How to convert an object's location to another's rotation
What you need to do is to copy the distance (or absolute position) of the "belt" to the cylinder's rotation, and if necessary, multiply by a factor. To achieve this you can use Animation Drivers.

To do this, select the object you want to rotate, press N and right click over the Z rotation filed and select Add Single Drivers (or if you want to select Add Drivers to add the drivers to the three of them).



Then, go to the Graph Editor (Ctrl + Left Arrow twice, from the default view), change the type to Drivers, like so:



Then select the Driver on the left, press N, scroll down and on the Drivers part, and use this settings:



Where the plane is the "belt"; If you need to, in the Expression field you can change the factor to suit your needs.